Quantum inverse scattering for the 20-vertex model up to Dynkin automorphism: crossing probabilities, 3D Poisson structure, triangular height functions, weak integrability

Abstract: We initiate a novel application of the quantum-inverse scattering method for the 20-vertex model, building upon seminal work from Faddeev and Takhtajan on the study of Hamiltonian systems, with applications to crossing probabilities, 3D Poisson structure, triangular height functions, and integrability. In comparison to a previous work of the author in late $2023$ which characterized integrability of a Hamiltonian flow for the 6-vertex model from integrability of inhomogeneous limit shapes, formalized in a work of Keating, Reshetikhin and Sridhar, notions similar to those of integrability can be realized for the 20-vertex model by studying new classes of higher-dimensional L-operators. In comparison to two-dimensional L-operators expressed in terms of Pauli basis elements, three-dimensional L-operators provided by Boos and colleagues have algebraic, combinatorial, and geometric, qualities, all of which impact leading order approximations of correlations, products of L-operators, the transfer matrix, and the quantum monodromy matrix in finite volume. In comparison to the inhomogeneous 6-vertex model, the 20-vertex model does not enjoy as strong of an integrability property through the existence of suitable action-angle variables, which is of interest to further explore, possibly from information on limit shapes given solutions to the three-dimensional Euler-Lagrange equations.